Forum Geometricorum Volume 14 (2014) 1–13. FORUM GEOM ISSN 1534-1178 Angle and Circle Characterizations of Tangential Quadrilaterals Martin Josefsson Abstract. We prove five necessary and sufficient conditions for a convex quadri- lateral to have an incircle that concerns angles or circles. 1. Introduction A tangential quadrilateral is a convex quadrilateral with an incircle, i.e., a circle inside the quadrilateral that is tangent to all four sides. In [4] and [5] we reviewed and proved a total of 20 different necessary and sufficient conditions for a convex quadrilateral to be tangential. Of these there were 14 dealing with different dis- tances (sides, line segments, radii, altitudes), four were about circles (excluding their radii), and only two were about angles. In this paper we will prove five more such characterizations concerning angles and circles. First we review two that can be found elsewhere. A characterization involving the four angles and all four sides of a quadrilateral appeared as part of a proof of an inverse altitude characterization of tangential quadrilaterals in [6, p.115]. According to it, a convex quadrilateral ABCD with sides a = AB, b = BC , c = CD and d = DA is tangential if and only if a sin A sin B + c sin C sin D = b sin B sin C + d sin D sin A. In the extensive monograph [9, p.133] on quadrilateral geometry, the following characterization is attributed to Simionescu. A convex quadrilateral is tangential if and only if its consecutive sides a, b, c, d and diagonals p, q satisfies |ac - bd| = pq cos θ where θ is the acute angle between the diagonals. The proof is a simple application of the quite well known identity 2pq cos θ =   b 2 + d 2 - a 2 - c 2   that holds in all convex quadrilaterals. Rewriting it as 2pq cos θ =   (b + d) 2 - (a + c) 2 + 2(ac - bd)   , we see that Simionescu’s theorem is equivalent to Pitot’s theorem a + c = b + d for tangential quadrilaterals. In Theorem 2 we will prove another characterization for the angle between the diagonals, but it only involves four different distances instead of six. Publication Date: January 23, 2014. Communicating Editor: Paul Yiu.